Thinning of Renewal Process. By Nan-Cheng Su. Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, Taiwan, 804, R.O.C.

Transcription

1 Thinning of Renewal Process By Nan-Cheng Su Department of Applied Mathematics National Sun Yat-sen University Kaohsiung, Taiwan, 804, R.O.C. June 2001

2 Contents Abstract ii 1. Introduction Preliminaries Multinomial Thinning Generalization of Multinomial Thinning References i

3 Abstract In this thesis we investigate thinning of the renewal process. After multinomial thinning from a renewal process A, we obtain the k thinned processes, A i, i = 1,, k. Based on some characterizations of the Poisson process as a renewal process, we give another characterizations of the Poisson process from some relations of expectation, variance, covariance, residual life of the k thinned processes. Secondly, we consider that at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the ith new atom assigned to the process D i, i = 1,, k. We also have characterizations of the Poisson process from some relations of expectation, variance of the process D i, i = 1,, k. Key words: characterization, exponential distribution, Poisson distribution, Poisson process, renewal process, residual life, thinning process ii

4 Thinning of Renewal Process 1 1 Introduction In this thesis we investigate thinning of the renewal process. Let {X n 0, n 1} be a sequence of independent and identically distributed (i.i.d.) positive random variables with the common distribution function F. F is called life distribution and we assume F (0) = 0 n throughtout this thesis. Let S 0 = 0, S n = X i, n 1. Also let A(t) = sup{n S n t}, t 0. Then A {A(t), t 0} is called a renewal process, X n is called the n-th inter-arrival time of A and the sequence {S n, n 1} is called arrival times of A. The random variables δ t = t S A(t), γ t = S A(t)+1 t, and β t = γ t +δ t, will be called respectively current life at t, residual life at t, and total life at t of the renewal process A. In the above set up, if {X n 0, n 1} are still assumed to be independent, but only X 2, X 3, are identically distributed with the common distribution function F, while X 1 has possibly a different distribution function, say H with H(0) = 0, then {X n 0, n 1} form the inter-arrival times of a delayed renewal process, say A d = {A d (t), t 0}. In particular, if i=1 H(t) = t 0 1 F (u)du E(X 2 ) holds, then the process becomes a stationary renewal process, say A s = {A s (t), t 0}. When F is exponentially distributed, then A becomes a Poisson process. Poisson processes are related to many distributions. The most important ones are uniform, exponential and Poisson distributions. Many interesting properties of the Poisson process are more or less based on the characteristics of these three distributions. For example, it is known that given A(t) = n, S 1 S 2 S n are distributed as the order statistics of n i.i.d random variables with the common uniform distribution on [0, t]. Huang et al. (1993) prove that given a function G, under mild conditions, as long as E(G(δ t ) A(t) = n) = E(G(X 1 ) A(t) = n), t 0, (1)

5 Thinning of Renewal Process 2 holds for a single positive integer n, then A is a Poisson process. Li et al. (1994) and Huang and Su (1997) prove that for some fixed n and k, k n, if E(S r k A(t) = n) = at r and E(S s k A(t) = n) = bt s, t 0, (2) for certain pairs of r and s, where a and b are independent of t, then A has to be a Poisson process. Note that the fact that (1) and (2) hold for A to be a Poisson process are based on the so called order statistics property. When A is a Poisson process, not only X n, but also γ t, independent of t, is exponentially distributed. This is resulted from the memoryless property of the exponential distribution. Gupta and Gupta (1986) prove that given a function G, under certain conditions, if E(G(γ t )) = c, t 0, where c > 0 is a constant, then A is a Poisson process. If Var(γ t ) is a constant and F is assumed to be continuous, Huang and Li (1993) prove that F is the exponential distribution function. Obviously, when A is a Poisson process, it is true that Var(γ t ) = E 2 (γ t ), t 0. Conversely, Huang and Chang (2000) prove that the equality Var(γ t ) = E 2 (γ t ), t 0, implies A is a Poisson process. Poisson process also has the superposition property. That is the superposition of independent Poisson processes is still a Poisson process. Conversely, if A is a Poisson process, independent thinned processes can be obtained through thinning operations on A. Based on the above Poisson characteristic, Chandramohan et al. (1985), Chandramohan and Liang (1985) and Huang (1989) prove that the existence of a pair of uncorrelated thinned processes implies the renewal process is Poisson. As a by-product, Chandramohan et al. (1985) prove that for a stationary renewal process A s, Var(A s (t)) = E(A s (t)), t 0, holds if and only if A s is a Poisson process. A similar result for an ordinary renewal process is given by Chen (1994). Note that mean and variance are equal is also an interesting

6 Thinning of Renewal Process 3 property enjoyed by the Poisson random variable. Furthermore, we introduce some important results about thinning of point process. Let N be a point process on R +. Then we obtain a thinned point process N p by retaining every point of the process with a certain constant probability p and deleting it with probability 1 p, independently of all other points and independently of the point process. N p is called the p-thinning of N, N is the p-inverse of N p and q = 1 p is the thinning probability. The p-inverse of any thinned point process is unique in distributional meaning (cf. Thedéen, 1986) and it will be also called the original process. Both the Cox (also called doubly stochastic Poisson) processes and the renewal processes are natural generalizations of Poisson processes. There is a basic characterization of Cox processes due to Mecke (1968). It says that a point process is a Cox process if and only if, for every p in (0, 1), there exists a point process which is the p-inverse of the process. Yannaros (1988b) gives some necessary and sufficient conditions for an ordinary or delayed gamma renewal process to be Cox. When F is gamma distributed, renewal process is called gamma renewal process. Yannaros proves that an ordinary gamma renewal process is a Cox process if and only if 0 < α 1, where α is a shape parameter of gamma distribution. For definitions and properties of Cox processes see Grandell (1976) or Karr (1986). Yannaros (1989) gives the following result. If a renewal process is a Cox process, then it cannot be the thinning of some non-renewal process, and all the possible original processes are p-thinnings of other renewal processes for every thinning parameter p (0, 1). This property characterizes the processes which are both Cox and renewal. Yannaros (1988a) proves that a renewal process is not the p-thinning of some non-renewal process for any p < 1, and the inverse of a thinned renewal process has to be renewal. In this thesis based on the above results, we present some characterizations of the Poisson process from thinning of renewal process.

7 Thinning of Renewal Process 4 2 Preliminaries Let M(t) = E(A(t)), and supp(g) denote the support of the function G. The following result is proved by Huang and Chang (2000). Theorem 1. Let E(X 1 ) <, M(t) = E(A(t)) be continuous with inf supp (M) = 0. Also assume M +(0) exists and that Var(A(t)) = be(a(t)), t 0, where b is a constant. Then b = 1 and F is exponential. The next theorem is for the stationary renewal process. It is also proved by Huang and Chang (2000). Note that it is implicitly assumed that the common inter-arrival time distribution function F is not periodic for a stationary renewal process. Theorem 2. Let Var(A s (t)) = be(a s (t)), t 0, where b is a constant. Then b = 1 and {A s (t), t 0}, is a Poisson process. As mentioned in Section 1, Huang and Li (1993) prove the following theorem which is a characterization of the Poisson process. Theorem 3. Assume F is an absolutely continuous distribution function with density function F (t), t 0. Then Var(γ(t)) is independent of t, if and only if F is exponentially distributed.

8 Thinning of Renewal Process 5 The following lemma has been proved by Huang and Li (1993), which gives an expression for the covariance of γ i (t) and γ j (t). Lemma 1. Assume Var(X 1 ) <. For every i j and t 0, Cov(γ i (t), γ j (t)) = Var(γ(t)) E 2 (X 1 ) + 1 p i p j p i + p j (Var(X 1 ) E 2 (X 1 )). (3) 3 Multinomial Thinning Now let {ξ n, n 1} be a sequence of i.i.d. random variables with distribution given k by P (ξ n = i) = p i, i = 1,, k, where 0 < p i < 1 and p i = 1. Thus we obtain the k thinned processes, {A i (t), t 0}, i = 1,, k, as follows: A i (t) = I (Sn t)i (ξn=i), n=1 i=1 by thinning the renewal process A through {ξ n, n 1}. On the other hand, the n-th point of A is assigned to the process A i, i = 1,, k, respectively, according to ξ n = i,i = 1,, k. Obviously, E(A i (t)) = p i p 1 j E(A j (t)), i j, i.e. E(A i (t)) is proportional to E(A j (t)), t 0, i j, for every renewal process A. For the case k = 2, Huang and Chang (2000) stated and proved that if Var(A 1 (t)) = cvar(a 2 (t)), t 0, where c is a constant, then A is a Poisson process and c = p 1 /(1 p 1 ). Based on Theorem 1, we investigate characterizations of A to be a Poisson process from the relationship between variances or expectations of A i and A j, i j. We give the general case in the following. Theorem 4. If, for some fixed i, j, i j, Var(A i (t)) = cvar(a j (t)), t 0, (4)

9 Thinning of Renewal Process 6 where c is a constant, then A is a Poisson process and c = p i /p j. Proof. The following arguments are very similar to the proof of Corollary 1 of Huang and Chang (2000). We know that the conditional distribution of A i (t) given A(t) is Binomial(A(t), p i ), i = 1,, k. First it is trivial to obtain Var(A i (t)) = p i (1 p i )E(A(t)) + p 2 i Var(A(t)), Var(A j (t)) = p j (1 p j )E(A(t)) + p 2 jvar(a(t)). Substituting these into (4), yields Var(A(t)) = cp j(1 p j ) p i (1 p i ) E(A(t)). p 2 i cp 2 j The assertions then follow from Theorem 1 immediately. The following consequence is immediate. Hence we omit the prove. As expected, there is a similar corollary based on Theorem 2 for the case of stationary renewal process. We also omit the statement of this result. Corollary 1. If, for some fixed i, j, i j, Var(A i (t)) = ce(a j (t)), t 0, where c is a constant, then A is a Poisson process and c = p i /p j. As mentioned in Section 1, Chandramohan et al. (1985), Chandramohan and Liang (1985) and Huang (1989) investigate characterizations of A to be a Poisson process from uncorrelatedness of A i and A j, for some i j. Let γ i (t) denote the residual life at time t of the renewal process A i, i = 1,, k. From a different direction Huang and Li (1993) prove

10 Thinning of Renewal Process 7 that if {A(t), t 0} is a Poisson process, then for every i j, Cov(γ i (t), γ j (t)) = 0, t 0. Conversely, Huang and Li (1993) also have the result that if there exist i j such that Cov(γ i (t), γ j (t)) = 0, t 0, then {A(t), t 0} is a Poisson process. Based on Theorem 3 and Lemma 1 we are now ready to state and prove the next corollary. Corollary 2. For some fixed i, j, m, n, where (i, j) (m, n). Let Cov(γ i (t), γ j (t)) 0. Assume, for some constant b 0, Cov(γ i (t), γ j (t)) = bcov(γ m (t), γ n (t)), t 0. (5) Then (i) b 1 implies A is a Poisson process. (ii) b = 1 implies p i + p j = p m + p n. Proof. Based on Lemma 1, we substitute (3) in (5) and obtain Var(γ(t)) = E 2 (X 1 ) + (1 + b )(Var(X 1 ) E 2 (X 1 )), 1 b p m + p n 1 b p i + p j where b 1. As Var(γ(t)) equals to a constant, from Theorem 3 we obtain A is a Poisson process. This proves part (i). Now assume b = 1, we substitute (3) into (5) again. After some simplifications we obtain 1 p i p j p i + p j = 1 p m p n p m + p n, and part (ii) follows. Now we want to see how to use Var(γ i (t)) to characterize the process. The following lemma gives an expression for the variance of γ i (t). Lemma 2. Assume Var(X 1 ) <. Then for every i and t 0, Var(γ i (t)) = Var(γ(t)) + 1 p i p i Var(X 1 ) + 1 p i E 2 (X p 2 1 ). (6) i

11 Thinning of Renewal Process 8 Proof. The following arguments are similar to those of Lemma 1. First, for convenience we define It is easy to see h I (ξr i) = 1 and r=k h X m = 0 if k > h. m=k γ i (t) = From (7) we obtain h=a(t)+1 (γ(t) + S h S A(t)+1 )I (ξh =i) h 1 r=a(t)+1 I (ξr i). (7) We also obtain γ 2 i (t) = = = = = h=a(t)+1 h=a(t)+1 +2 E[γ i (t)] = E[γ(t)] + 1 p i p i E[X 1 ]. (8) (γ(t) + S h S A(t)+1 )I (ξh =i) (γ(t) + S h S A(t)+1 )I (ξh =i) h=a(t)+1 l=h+1 (γ(t) + S l S A(t)+1 )I (ξl =i) h=a(t)+1 h=a(t)+1 h=a(t)+1 γ(t) + γ 2 (t) + 2γ(t) γ 2 (t) + 2γ(t) I (ξh =i) h 1 I (ξr i). r=a(t)+1 Here we have used the fact l 1 r=a(t)+1 h 1 r=a(t)+1 h 1 r=a(t)+1 (γ(t) + S h S A(t)+1 )I (ξh =i) 2 I (ξr i) 2 I (ξr i) h 1 r=a(t)+1 I (ξr i) l 1 I (ξr i) r=a(t)+1 2 h h 1 X m I (ξh =i) I (ξr i) m=a(t)+2 r=a(t)+1 2 h h h 1 X m + X m I(ξh=i) I (ξr i) m=a(t)+2 m=a(t)+2 r=a(t)+1 h h h 1 h I (ξr i) = m=a(t)+2 X m + m=a(t)+2 X 2 m + 2 h 1 l 1 I (ξr i)i (ξh i) r=a(t)+1 s=h+1 m=a(t)+2 n=m+1 I (ξs i). X m X n

12 Thinning of Renewal Process 9 Taking the expectation of γi 2 (t), we obtain E[γi 2 (t)] = p i (1 p i ) h 1 A(t) E[γ 2 (t)] + 2E[γ(t)]E[ = h=a(t)+1 +E[ h m=a(t)+2 h=a(t)+1 X 2 m] + 2E[ h 1 h m=a(t)+2 n=m+1 X m X n ] h m=a(t)+2 X m ] p i (1 p i ) h 1 A(t) ( E[γ 2 (t)] + 2(h A(t) 1)E[γ(t)]E[X 1 ] +(h A(t) 1)E[X 2 1] + (h A(t) 2)(h A(t) 1)E 2 [X 1 ] ) = p i (1 p i ) ( h E[γ 2 (t)] + 2hE[γ(t)]E[X 1 ] + he[x1] 2 + h(h 1)E 2 [X 1 ] ) h=0 = p i (1 p i ) ( h E[γ 2 (t)] + 2hE[γ(t)]E[X 1 ] + hvar(x 1 ) + h 2 E 2 [X 1 ] ) h=0 = p i E[γ 2 (t)] (1 p i ) h + 2p i E[γ(t)]E[X 1 ] h(1 p i ) h h=0 h=0 +p i Var(X 1 ) h(1 p i ) h + p i E 2 [X 1 ] h 2 (1 p i ) h h=0 h=0 = E[γ 2 (t)] + 2(1 p i) E[γ(t)]E[X 1 ] + 1 p i Var(X 1 ) p i p i + (2 p i)(1 p i ) E 2 [X p 2 1 ]. (9) i Finally, it follows from (8) and (9) that as desired. Var(γ i (t)) = E[γ 2 i (t)] E 2 [γ i (t)] = Var(γ(t)) + 1 p i p i Var(X 1 ) + 1 p i E 2 (X p 2 1 ), i Having given Theorem 3 and Lemma 2 we are now ready to state and prove the next corollary. Corollary 3. For some fixed i, j, i j. Assume, for some constant b > 0, Var(γ i (t)) = bvar(γ j (t)), t 0. (10)

13 Thinning of Renewal Process 10 Then (i) b 1 implies A is a Poisson process. (ii) b = 1 implies p i = p j. Proof. By using (6), (10) becomes Var(γ(t)) = ( b 1 )Var(X 1 ) b p i 1 b p j 1 b ( 1 b 1 p i p j p 2 i + b )E 2 (X p 2 1 ), j where b 1. Again Theorem 3 implies part (i). Now assume b = 1, substituting (6) in (10) and after some simplifications, we obtain p i p j Var(X 1 ) + (p i p j )(p i p i p j + p j ) E 2 (X p i p j p 2 i p 2 1 ) = 0. j By using the fact that Var(X 1 ) > 0 and E 2 (X 1 ) > 0, p i = p j is the only possibility. This proves part (ii). p j p 1 i Note that when A is a Poisson process, Var(γ i (t)) = p 2 jp 2 i Var(γ j (t)), E(γ i (t)) = E(γ j (t)) and Var(γ i (t)) = p 2 jp 2 E 2 (γ j (t)), i j, hold. Next we give another consequence. Again the proof is omitted. i Corollary 4. For some fixed i. Assume, for some constant b > 0, Var(γ i (t)) = bvar(γ(t)), t 0. Then (i) b 1 implies A is a Poisson process. (ii) b = 1 implies p i = 1. 4 Generalization of Multinomial Thinning Consider a renewal process A {A(t), t 0} defined on a probability space (Ω, F, P ), where it is assumed that (i) the state space consists of non-negative integers, (ii) A(0) =

14 Thinning of Renewal Process 11 0, A(t) <, a.s., and (iii) A is a separable point process with non-decreasing rightcontinuous sample paths with unit step jumps. Let {T n, n 1} be the sequence of arrival times of A. Assume there are X n arrivals at time T n, where {X n, n 1} are i.i.d. random variables with the same distribution as some random variable, say X, which only takes on positive integers. For each arrival, independent of everything else, split the mass of this unit atom into k new atoms according to β, where β = (β 1, β 2,, β k ) is a random vector in R k such that P (β 1 + β β k = 1) = 1, so that for i = 1, 2,, k, the ith new atom has (possibly negative) mass β i and is assigned to the process D i {D i (t), t 0}. Thus D i (t) can be represented as D i (t) = A(t) X n n=1 m=1 β imn, (11) where { β mn }, with β mn = (β 1mn, β 2mn,, β kmn ), is a two-dimensional array of i.i.d. random vectors with the same distribution as β. Again let E k denote the set {e 1,, e k }, where e i is the element of R k with a 1 in the ith position and zeros elsewhere. The case in section 2 is for β E k. Let E(X) = µ, E(β i ) = p i, i = 1,, k, E(β 2 i ) = γ ii, i = 1,, k. Assume p i <, γ ii < and 1 E(X 2 ) <. Huang (1989) gives an expression for the pointwise covariance of D i and D j. For completeness we supply a proof here. Lemma 3. Var(D i (t)) = γ ii µe[a(t)] + p 2 i E[X(X 1)]E[A(t)]

15 Thinning of Renewal Process 12 +p 2 i µ 2 E[A(t)(A(t) 1)] (p i µe[a(t)]) 2 (12) = p 2 i µ 2 Var(A(t)) + [p 2 i Var(X) p 2 i µ + γ ii µ]e[a(t)]. (13) Proof. First it is easy to obtain E[D i (t)] = p i µe[a(t)]. (14) Now E[Di 2 (t)] A(t) = E A(t) = E 2 X n β imn n=1 m=1 X n n=1 m=1 β 2 imn + A(t) X n n=1 m r β imn β irn + A(t) n s ( Xn ) ( Xs β imn m=1 r=1 β irs ) = µγ ii E[A(t)] + p 2 i E[X(X 1)]E[A(t)] + p 2 i µ 2 E[A(t)(A(t) 1)], (15) where we have used the facts that E[β 2 imn] = E[β 2 i ] = γ ii, E[β imn β irs ] = E[β i ]E[β i ] = p 2 i, (m, n) (r, s), and for n s, E[X n X s ] = (E[X]) 2 = µ 2. Finally, the assertion follows easily from (14) and (15). Obviously, E(D i (t)) = p i p 1 j E(D j (t)), for some 1 i, j k, i j, i.e. E(D i (t)) is proportional to E(D j (t)), t 0, for every renewal process A. Yet given that Var(D i (t)) is proportional to Var(D j (t)), t 0, will imply A is Poisson. We state and prove the consequence of Theorem 1 in the following.

16 Thinning of Renewal Process 13 Corollary 5. If, for some fixed i, j, i j, Var(D i (t)) = cvar(d j (t)), t 0, (16) where c is a constant, then A is a Poisson process and c = γ iiµ + p 2 i (µ 2 µ + Var(X)) γ jj µ + p 2 j(µ 2 µ + Var(X)). Proof. From (12), it is easy to get Var(D i (t)) = p 2 i µ 2 Var(A(t)) + [p 2 i Var(X) p 2 i µ + r ii µ]e(a(t)), Var(D j (t)) = p 2 jµ 2 Var(A(t)) + [p 2 jvar(x) p 2 jµ + r jj µ]e(a(t)). Substituting these into (16), yields ( c(p 2 Var(A(t)) = j Var(X) p 2 jµ + γ jj µ) (p 2 i Var(X) p 2 ) i µ + γ ii µ) E(A(t)). p 2 i µ 2 cp 2 jµ 2 The assertions now follow from Theorem 1 immediately. By using Theorem 1, it can be shown that if Var(D i (t)) is proportional to E(D j (t)), t 0, then A is Poisson process. Corollary 6. If, for some fixed i, j, i j, Var(D i (t)) = ce(d j (t)), t 0, where c is a constant, then A is a Poisson process and c = p2 i µ 2 p 2 i µ + γ ii µ + p 2 i Var(X). p j µ In the following we consider the case of stationary renewal process. Again we assume

17 Thinning of Renewal Process 14 that the common interarrival time distribution function F is not periodic for a stationary renewal process. We give D si (t) = A s(t) X n n=1 m=1 β imn, instead of (11) for the stationary renewal process. We can obtain the following result. Let M s (t) = E(A s (t)) and M k is the k-fold convolution of M with itself. Theorem 5. If, for some fixed i, j, i j, Var(D si (t)) = cvar(d sj (t)), t 0, (17) where c is a constant, then A s is a Poisson process and c = γ iiµ + p 2 i E[X(X 1)] γ jj µ + p 2 je[x(x 1)]. Proof. From Lemma 3, it is easy to obtain Var(D si (t)) = µγ ii E[A s (t)] + p 2 i E[X(X 1)]E[A s (t)] Again for every positive integer k, the identity +p 2 i µ 2 E[A s (t)(a s (t) 1)] (p i µe[a s (t)]) 2. E(A s (t)(a s (t) 1) (A s (t) k + 1)) = k!m s (t) M k 1 (t), t 0, see Lemma 2.1 of Chandramohan et al. (1985), we have E(A s (t)(a s (t) 1)) = 2 t M(t x)dx = 2 t M(x)dx, η 1 η 1 0 where again η 1 = E(X 2 ). We also know that E(A s (t)) = t η 1. Substituting these into (17), yields t µγ ii + p 2 i E[X(X 1)] t + p 2 i µ 2 2 t η 1 η 1 η 1 0 t = cµγ jj + cp 2 η je[x(x 1)] t + cp 2 1 η jµ η 1 0 M(x)dx (p i µ t ) 2 η 1 t 0 M(x)dx c(p j µ t η 1 ) 2.

18 Thinning of Renewal Process 15 Taking the derivatives of both sides with respect to t and simplifying, we obtain that M(t) = c(γ jjµ + p 2 je[x(x 1)]) (γ ii µ + p 2 i E[X(X 1)]) 2µ 2 (p 2 i cp 2 j) Finally letting t = 0, we obtain c = γ iiµ + p 2 i E[X(X 1)] γ jj µ + p 2 je[x(x 1)], and M(t) = t η 1 follows. Consequently F is exponential as required. + t η 1. Note that the above theorem can also be proved by using Theorem 3. For the stationary case, E(D si (t)) = p i p 1 j E(D sj (t)) still holds as in the ordinary case. Similar remark can be applied for Theorem 5 and the following Corollary 7. Corollary 7. Let, for some fixed i, j, i j, Var(D si (t)) = ce(d sj (t)), t 0, where c is a constant. Then {A s (t), t 0} is a Poisson process and c = γ iiµ + p 2 i E[X(X 1)]. p j µ

19 Thinning of Renewal Process 16 References 1. Chandramohan J, Foley RD, Disney RL (1985) Thinning of point processes-covariance analysis. Adv. Appl. Prob. 17, Chandramohan J, Liang LK (1985) Bernoulli, multinomial and Markov chain thinning of some point processes and some results about the superposition of dependent renewal processes. J. Appl. Prob. 22, Chen YH (1994) Classes of life distributions and renewal counting processes. J. Appl. Prob. 23, Grandell J (1976) Doubly Stochastic Poisson Processes. Lect. Notes Math. 529 Springer-Verlag, New York. 5. Gupta PL, Gupta RC (1986) A characterization of the Poisson process. J. Appl. Prob. 23, Huang WJ (1989) On some characterizations of the Poisson process. J. Appl. Prob. 26, Huang WJ, Li SH (1993) Characterizations of the Poisson process using the variance. Commun. Statist.-Theory Meth. 22, Huang WJ, Li SH, Su JC (1993) Some characterizations of the Poisson process and geometric renewal process. J. Appl. Prob. 30, Huang WJ, Su JC (1997) On a study of renewal process connected with certain conditional moments. Sankhyā Ser. A 59, Huang WJ, Chang WC (2000) On a study of the exponential and Poisson characteristics of the Poisson process. Metrika 50,

20 Thinning of Renewal Process Karr AF (1986) Point Processes and Their Statistical Inference. Marcel Dekker, New York. 12. Li SH, Huang WJ, Huang MNL (1994) Characterizations of the Poisson process as a renewal process via two conditional moments. Ann. Inst. Stat. Math. 46, Mecke J (1968) Eine charakteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrschein. und Verw. Geb. 11, Thedéen T (1986) The inverses of thinned point processes. Dept. of Statistics, University of Stockholm, Research Report 1986: Yannaros N (1988a) The Inverses of Thinned Renewal Processes. J. Appl. Prob. 25, Yannaros N (1988b) On Cox Processes and Gamma Renewal Processes. J. Appl. Prob. 25, Yannaros N (1989) On Cox and Renewal Processes. Stat. Prob. Lett. 7,